No load open secondary, bad tranny, go for a replacement, running hot a replacement would make you feel a lot more comfortable in the long run. Unless the Manufacturer SAYS it is supposed to run at 45 C open secondary. I can not see why it would really, not under normal design parameters, maybe a 3 to 5 C rise over ambient yeah, Then again is your ambient around 41 C ? If so then the tranny may be fine!
Harlan

There are reasons actually that a tranny would get this warm
1. Shorts in the secondary windings, ie between layers, damaged insulation, etc.
2. shorted windings in the primary as well, same way.
3. A mistake in winding the transformer at the factory, we all have bad days and maybe the wire was supposed to be AWG14 and is AWG20??
4. A Bad core! I have seen where the laminations did not get a good oxide coating and the core although magnetically perfect, was a total failure due to eddy currents, and man did they get hot!, I was ready to start a new Induction welder with those.

So there are reasons all of which are most likely not in your design criteria, so I would say a replacement is best way to go.
Good luck too.
Harlan